A Sufra food bank volunteer in Kilburn, north London, hands over a bag of food to a woman and child. Photograph: Martin Godwin for the Guardian

The Sufra food bank, run from a small community centre on a north-west London housing estate, is in high demand. The Muslim charity's Wednesday afternoon session sees a constant stream of people coming to claim food packages: young couples, men on their own and mothers trying to keep toddlers quiet.

Ali Jawad, a 22-year-old business student and regular volunteer, takes some details from another Sufra "guest" and begins to fill blue plastic bags with cereal, soup cans, pasta, rice, biscuits and baby food. The amount given depends on the size of the family, but is designed to get each one through the next five days.

"It's been a real eye-opener," says Jawad. "You know poverty is out there, but you don't always see it. So to speak to single mothers who can't afford food for their child, or someone walking miles across London to get here, it can be heartbreaking. We try to be a friendly face as well as providing the food they need."

Food banks in the UK are overwhelmingly operated by church-affiliated groups, including the 421 outlets run by Christian charity, the Trussell Trust. Yet Sufra, meaning "come to the table" in Arabic, is one of a growing number of Muslim organisations also attempting to tackle food poverty.

Like many Christians motivated by their faith to help others, giving to the less fortunate is an important tenet of Islam. Voluntary donations and charity work (Sadaqah) among British Muslims have typically been geared towards poverty relief projects overseas. But Sufra organisers say many Muslims now "increasingly feel they have a responsibility to the wider community and the problems here in Britain".

Sufra and parent charity Al-Mizan (which provides grants of up to £500 to individuals in crisis across the UK) offer support to anyone in need, regardless of religion or ethnic background. Halal and non-halal foods are available: cans of pork sausages are carefully marked out with red stickers.

Stephen Lashbrook is a single father living in south Kilburn with his baby daughter, who is now almost two. Out of work following a construction work injury in 2010, this is the fourth time he has been forced to turn to Sufra for help.

"It's been a real struggle this winter," he says. "Money has been very tight. I get £71 income support allowance each week, and it's been tough making that stretch. We've got a pay-as-you-go meter for the utilities so you can watch what you're spending, but during the cold months you can't go without the heating. Once the rent and council tax are paid, food is the thing you just can't find the money for."

It was Lashbrook's sister who told him about the food bank. "It didn't matter to me about the background of the volunteers. I didn't even think about it. These guys have been a big help."

Mohammed Sadiq Mamdani, director of the Al-Mizan, says demand has gradually grown since they started in October last year. Initially, Sufra's 12 volunteers typically handed out 20 food packages each week, but in the past three months the number has risen to around 40. Since most attending the food bank have families of four to five, Mamdani calculates that the charity has provided food for more than 3,000 people in its first six months.

Some of the interest is down to word spreading around the local referring agencies: GP surgeries, housing associations, refugee charities and Brent council. But Mamdani rejects the idea promoted by some Conservative politicians that people are simply taking advantage of free food. He says that benefit cuts, delays and sanctions are having an increasingly painful impact. Sufra organisers have tried, sometimes in vain, to limit each guest to four food package vouchers a year.

"Benefit sanctions might last for a month or two, which means for some people there's a crisis in their life that will be going on for many weeks," Mamdani explains. "So we say to the referring agencies, "If someone needs a fifth or sixth voucher, talk to us."

"We don't encourage dependency," he adds. "We are simply responding to the reality of food poverty. What we have now is a situation where welfare payments are token contributions to people's needs. So it's not a problem that's going away until some of the wider economic issues change."

Sufra will soon be moving to larger premises where it will launch a "food academy" to train vulnerable young people – care leavers, ex-offenders and those recovering from addiction – how to cook and manage budgets. "It would have been naive to think the food bank alone is enough," Mamdani explains.

Muslim-run food aid projects across Britain have emerged from seasonal events held during Ramadan: mosques or cultural groups inviting homeless people to share in Iftar feasts. Muslim students organising on social media have dubbed the meals "flashmob Iftars".

In 2009, the Islamic Society of Britain started one-off eat 'n' meet events along similar lines. Last November, the society began helping fund a weekly Peace Centre food bank in a deprived part of Leicester.

"More than 90% of the people coming along are white, non-Muslim," says project co-ordinator Salma Ravat. "Some are upset, some are putting a brave face on it. It's not always easy for them to get past the stigma of using a food bank, but with benefit cuts, they are struggling."

Ravat believes this reflects the growing involvement of Muslims in "mainstream" voluntary work. "Increasingly, Muslims want to give time and energy to the community around them," he says.

Many mosques have acted as donation points for food going to local charities elsewhere, but Birmingham Central mosque now runs its own small food bank, distributing parcels to a handful of people each Saturday. "We've seen a significant increase in the last few years of needy people asking for basic necessities like food," says mosque administrator Muhammad Ali. "The poverty is rising for all sorts of reasons, sadly, and the Muslim community is part of those struggles."

In Blackburn, a food donation drive organised by One Voice Blackburn during Ramadan last year led to five mosques permanently hosting collection bins for the town's Trussell Trust food bank. "It's really encouraging that all communities are now part of the process," says Ros Duerden, chair of Blackburn Food bank.

Back in London, since February 2013, the Muslim Association of Croydon has organised a Friday evening "feed the homeless" soup kitchen on the forecourt of a hostel near the Croydon mosque. But many of the people queuing up for cups of lentil soup and chicken biriyani cooked by a local restaurant are not homeless.

"We've had people of all backgrounds and situations: some in emergency accommodation and some struggling to afford the rent. A few have said they are in work, part-time or on zero-hour contracts," says the group's general secretary Ashtaq Arain." They're still struggling to get hot food and pay all the bills so they stop by here."

Arain points to a younger volunteer. "Charity is very important to our faith, and second and third generation British Muslims are now beginning to take up voluntary work more actively. I have great hopes for the next generation."